Method for suppressing clutter in space-time adaptive processing systems

ABSTRACT

A method surpresses clutter in a space-time adaptive processing system. The method achieves low-complexity computation via two steps. First, the method utilizes an improved fast approximated power iteration method to compress the data into a much smaller subspace. To further reduce the computational complexity, a progressive singular value decomposition (SVD) approach is employed to update the inverse of the covariance matrix of the compressed data. As a result, the proposed low-complexity STAP procedure can achieve near-optimal performance with order-of-magnitude computational complexity reduction as compared to the conventional STAP procedure.

FIELD OF THE INVENTION

This invention relates generally to phased array radar systems, and moreparticularly to suppression clutter while detecting targets.

BACKGROUND OF THE INVENTION

A typical prior art space-time adaptive processing (STAP) systemincludes an array of N transmit and receive antennas. The antenna arraycan be mounted on a moving platform, e.g., a plane or a boat, to locateair, ground and sea targets. STAP systems are also used bymeteorologists and geologists.

The receiver antenna gain pattern can be steered in a desired directionby a beam forming process. Advanced STAP systems are required to detecttargets in the presence of both clutter and jamming. Ground or seaclutter is extended in both angle and range, and is spread in Dopplerfrequency because of the platform motion.

The STAP system uses a pulse train, and coherent pulse integration. Acoherent processing interval (CPI) defines the duration of the pulsetrain. During each CPI, the transmitter sends out M pulses using thetransmit antennas. The time between the beginning of a pulse and thebeginning of the next pulse is called a pulse repetition interval (PRI).The pulses reflect from targets at different distances from the STAPsystem.

The range to a target is determined by the time interval between thesending of a pulse and receiving the reflected signal. The STAP systemcollects the reflected signals for each antenna, or each pulse andrange. The data derived from the reflected signals can be assembled intoa three-dimensional matrix, which is sometimes called a STAP cube.

The problem solved by the invention is shown schematically in FIG. 1. Insensing applications built on a moving platform, returned signals arecommonly contaminated by clutter returns in the form of interference101, which decreases the signal-to-noise ratio (SNR) from differentincoming angles 102 and Doppler frequencies 103. To accurately locatemoving targets, effective clutter suppression techniques areindispensable.

Among the many known clutter-suppression techniques, space-time adaptiveprocessing (STAP) is the most promising. In STAP, returned signals arefiltered simultaneously over space and time domains. As a result,clutter interference can be effectively suppressed regardless of theincoming angle and Doppler frequency.

However, the conventional STAP is handicapped by a prohibitivecomputational complexity. For M pulses and N antennas, the conventionalSTAP requires an intensive matrix inversion of dimension MN×MN. Forpractical systems with MN on the order of hundreds, such a large matrixinversion requirement makes it difficult to implement STAP for real-timetarget detection.

To circumvent this obstacle, considerable research efforts have beendevoted to developing low-complexity STAP. According to Brennan's rule,the rank of the clutter interference covariance matrix C is known to bemuch smaller than MN. Thus, one way to achieve complexity reduction isto compress the returned signal into an r-dimensional subspace withr<<MN. In particular, one low-complexity STAP exploits a subspacetracking called fast approximated power iteration (FAPI). FAPI can beemployed to effectively compress the returned signal into a much smallersignal subspace, which enables low-complexity STAP operating on thecompressed data.

SUMMARY OF THE INVENTION

The embodiments of the invention provide a two-step low-complexityspace-time adaptive processing (STAP) method. In the first step, we usea modified fast approximated power iteration (FAPI) procedure withimproved convergence before applying the modified FAPI procedure tocompress received signals.

In the second step, we use a progressive singular vector decomposition(PSVD)-based low-complexity technique to recursively determine theinverse of a covariance matrix of the compressed data to detect targets.

The resulting low-complexity STAP reduces the computational complexityof O((MN)³) for the conventional to O((MN)r).

The STAP according to embodiments of the invention achieves near-optimalperformance when compared to the conventional STAP using a full matrixinversion.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic of a clutter interference spectrum resolved byembodiments of the invention;

FIG. 2 is schematic of a return signal structure over range cellsaccording to embodiments of the invention;

FIG. 3 is a schematic of a low-complexity two-step STAP according toembodiments of the invention;

FIG. 4 is a block diagram of pseudo code for a fast modified poweriteration procedure according to embodiments of the invention; and

FIG. 5 is a block diagram of pseudo code for a progressive SVD accordingto embodiments of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The embodiments of our invention provide a method for detecting targetusing low complexity clutter suppression in space-time adaptiveprocessing (STAP) systems.

We use the following notational conventions. Vectors and matrices aredenoted with boldface, ∥•∥ represents the Euclidean norm of the enclosedvector, and |•| denotes the cardinality of the enclosed set. I_(N) isthe N×N identity matrix. We use (•)^(H) and R{•} for Hermitiantransposition and the real part, respectively. Finally, [A]_(i,j)denotes the i-th row and j-th column entry of matrix A, and A(:,j) isthe j-th column of the matrix A.

A sensing system transmits pulses of the forms(t)=R{A _(t) E(t)e ^(jω) ^(c) ^(t)},where ω_(c) is the carrier frequency, A_(t) and E(t) are the transmitpower and pulse waveform, respectively.

As shown in FIG. 2, the returned signals are arranged in K range cells201 for N antennas 202, and M pulses 203. The range cells correspond toa current time instant and previous time instants.

First, we construct a received signal data vector x(k) of length MN bystacking up samples collected over the M pulses 203 from each antenna202 in the k-th range cell 201, where k=1,2, . . . , K correspond totime instances.

To determine whether a target is present in the k-th range cell, aclutter-plus-noise covariance matrix C(k) of dimension MN×MN isdetermined from neighboring range cells, i.e., adjacent time instances,assuming that the neighboring range cells are impaired by the sameclutter, and yet target-free. An index Ω_(k) of the range cells is usedto determine C(k). Thus, C(k) can be expressed as

$\begin{matrix}{{C(k)} = {\frac{1}{{\Omega(k)}}{\sum\limits_{\ell \in {\Omega{(k)}}}\;{{x(\ell)}{{x(\ell)}^{H}.}}}}} & (1)\end{matrix}$

An optimal space-time filter for clutter suppression is given by C(k)⁻¹.Thus, the received signal x(k) is first filtered with C(k)⁻¹:z(k)=C(k)⁻¹ x(k).   (2)

Upon obtaining z(k), target detection can be performed. Despite the goodperformance of Eqn. (2), the matrix inversion C(k)⁻¹ for each range cellincurs prohibitive complexity, as described above.

To cope with this problem, different subspace-tracking procedures areknown to first reduce the dimension of x(k), before performing thematrix inversion.

We assume that a subspace concentration matrix W has dimensions MN×r,where rank(C(k))<r=MN. The compressed signal after subspaceconcentration process is given byy(k)=W ^(H) x(k),   (3)where W is given by the following optimization function

$\begin{matrix}{W = {\underset{\overset{\sim}{W}}{argmin}{{{{y(k)} - {\overset{\sim}{W}{\overset{\sim}{W}}^{H}{x(k)}}}}^{2}.}}} & (4)\end{matrix}$

The optimization problem in Eqn. (4) can be numerically solved usingsubspace-tracking procedures, such as FAPI. Next, the compressed signalis filtered withr(k)=R(k)⁻¹ y(k),   (5)where R(k) is the corresponding compressed clutter-plus-noise covariancematrix

$\begin{matrix}{{R(k)} = {\frac{1}{{\Omega(k)}}{\sum\limits_{\ell \in {\Omega{(k)}}}\;{{y(\ell)}{{y(\ell)}^{H}.}}}}} & (6)\end{matrix}$It is worth noting that R(k) is of dimension r×r, which is significantlysmaller than C(k). Finally, target detection is applied on thecompressed and filtered signal r(k).

FIG. 3 shows the two-step STAP according to the invention. Signalsreceived from the N antennas 301 are fed through delay lines with taps T302. The first step 310 performs a fast subspace concentration accordingto the invention. The second step performs the PSVD 320, which isfollowed by target detection 330. In the following, we describe thesesteps in details. Because the method is iterative, we use W(k) to denoteW derived from {x(l); l=1,2, . . . , k}.

Modified Fast Approximated Power Iteration (MFAPI)

The prior art FAPI procedure is an approximation of a projectionapproximation subspace tracking (PAST) procedure. By exploiting theapproximation of W(k)≈W(k−1), FAPI can reduce the computationalcomplexity of PAST from O(NMr²) to O(3NMr+5r²). However, the derivationof FAPI does not explicitly take into account the impact of additivenoise. As a result, the performance degrades as the signal-to-noiseratio (SNR) decreases.

More specifically, FAPI is derived from an approximated power iteration(API) procedure. In API, the auxiliary matrix Z is updated by

$\begin{matrix}{{{Z(k)} = {\frac{1}{\beta}{{\Theta(k)}^{H}\left\lbrack {I_{r} - {{g(k)}{y(k)}^{H}}} \right\rbrack}{Z\left( {k - 1} \right)}{\Theta(k)}^{- H}}},} & (7)\end{matrix}$whereΘ(k)=W(k−1)^(H) W(k),   (8)and g(k) is of length r.

It is important to observe that the last term Θ(k)^(−H) in Eqn. (7)incurs O(r³) operation but also may enhance noise if Θ(k) is noisy.Motivated by this observation, we provide the following twomodifications of Eqn. (7). Recalling that Θ is nearly orthonormal, it isreasonable to approximate Θ(k)^(−H) asΘ(k)^(−H)=Θ(k).   (9)

As a result, Eqn. (7) becomes

$\begin{matrix}{{Z(k)} = {\frac{1}{\beta}{{\Theta(k)}^{H}\left\lbrack {I_{r} - {{g(k)}{y(k)}^{H}}} \right\rbrack}{Z\left( {k - 1} \right)}{{\Theta(k)}.}}} & (10)\end{matrix}$

Note that Eqn. (10) has the same computational complexity as Eqn. (7).Further computation reduction can be achieved by observing that Wcomprises orthonormal column vectors. Hence, we can approximateΘ(k)^(−H)≈I_(r) in Eqn. (8) and Z(k) takes the following form:

$\begin{matrix}{{Z(k)} = {\frac{1}{\beta}{{\Theta(k)}^{H}\left\lbrack {I_{r} - {{g(k)}{y(k)}^{H}}} \right\rbrack}{{Z\left( {k - 1} \right)}.}}} & (11)\end{matrix}$

It should be pointed out that Eqn. (11) has O(r³) less operation ascompared to Eqns. (7) and (10).

We can re-derive our modified FAPI (MFAPI) by incorporating Eqn. (10)and Eqn. (11). The update functions for Z(k) using Eqns. (10) and (11)are given as follows, respectively.

$\begin{matrix}{{{Z(k)} = {\frac{1}{\beta}\left( {{Z\left( {k - 1} \right)} - {{g(k)}{h^{\prime}(k)}} - {{ɛ(k)}{g(k)}^{H}}} \right)}},{and}} & (12) \\{{{Z(k)} = {\frac{1}{\beta}\left( {{Z\left( {k - 1} \right)} - {{g(k)}{h^{\prime}(k)}}} \right)}},} & (13)\end{matrix}$where the definitions of h′(k) and ε(k) are shown in FIG. 4.

In an alternative embodiment, the MFAPI procedures employing Eqn. (12)and Eqn. (13) are referred to as the Noise-Robust MFAPI (NR-MFAPI) andLow Complexity MFAPI (LC-MFAPI). The total computational complexity ofNR-MFAPI and LC-MFAPI is O(3NMr+5r²) and O(3NMr+3r²), respectively.

Pseudo code for our NR/LC-MFAPI procedures is summarized in FIG. 4.

Progressive SVD (PSVD)

Despite the fact that the output of the subspace concentration, y(k),has a much smaller dimension as compared to x(k), computation of R(k)⁻¹in Eqn. (5) for k=1,2, . . . , K can still remain computationallyexpensive. To circumvent this obstacle, it is important to observe thatR(k) and R(k−1) are correlated. This is because they are derived fromsome common compressed data vectors and clutter variation betweenconsecutive pulse intervals is correlated.

Thus, we provide the PSVD approach by capitalizing on a thin SVDtechnique. More specifically, the PSVD approach determines R(k)⁻¹ interms of R(k−1)⁻¹ and ΔR(k)=R(k)−R(k−1), assuming R(k−1)⁻¹ is given andrank(ΔR(k))=r. Upon obtaining R(k)⁻¹, the same procedures can berepeated to derive ΔR(k+1)⁻¹ recursively.

The low-rank assumption of ΔR(k) allows us to decompose it into thefollowing form:

${{\Delta\;{R(k)}} = {\sum\limits_{d = 1}^{D_{k}}\;{\alpha_{d}q_{d}q_{d}^{H}}}},$where

$D_{k}\overset{def}{=}{{{rank}\left( {\Delta\;{R(k)}} \right)} = {r.}}$Furthermore, q_(d) and α_(d) are the eigenvectors and the associatedeigenvalues, respectively, with α₁≧α₂≧ . . . ≧α_(D) _(k) . To achievelow-complexity computation, we use the following rank-one approximationto decompose ΔR(k):ΔR(k)≈aa^(H).   (14)where a=√{square root over (α₁)}q₁.

Finally, we assume that R(1)⁻¹ is given and R(1) can be decomposed asR(1)=USU ^(H).   (15)

The pseudo code for the rank-one PSVD procedure is shown in FIG. 5.

Note that the computational complexity of the PSVD is O((3+2m)r²), ascompared to O(r³) for a direct conventional matrix inversion of R(k)⁻¹.To fully exploit the advantage of PSVD, we set (3+2m)=r. It should beemphasized that, rather than Eqn. (14), higher-rank approximation ofΔR(k) may lead to better approximation accuracy at the price of highercomputational complexity. As described below, the rank-one approximationin Eqn. (14) is usually sufficient to result in satisfactoryperformance.

Computational Complexity

The total computational complexity of our two-step STAP procedure isO(3NMr+(8+2m)r²), and O(3NMr+(5+2m)r²) for PSVD in conjunction withNR-MFAPI and LC-MFAPI, respectively.

Clearly, this stands for a substantial computational reduction ascompared to the full matrix inversion C(k)⁻¹ of O((NM)³) operation,particularly for practical values of N and M.

Effect of the Invention

The invention provides a two-step low-complexity space-time adaptiveprocessing (STAP) procedure for a sensing application mounted on amoving platform subject to strong clutters. The STAP procedure firstcompresses the received signals into a much smaller subspace using themodified FAPI procedure before recursively computing the inverse of thecovariance matrix of the compressed data using PSVD.

The resulting procedure has computational complexity ofO(3NMr+(8+2m)r²), and O(3NMr+(5+2m)r²) for the PSVD in conjunction withNR-MFAPI and LC-MFAPI, respectively.

This is an order-of-magnitude computational complexity reduction ascompared to the conventional STAP procedure that requires O((NM)³)operations.

Although the invention has been described by way of examples ofpreferred embodiments, it is to be understood that various otheradaptations and modifications may be made within the spirit and scope ofthe invention. Therefore, it is the object of the appended claims tocover all such variations and modifications as come within the truespirit and scope of the invention.

1. A method for detecting a target in a space-time adaptive processingsystem, comprising the steps of: receiving M signals, by N antennas,reflected from the target; passing the signals received by each antennathrough a delay line with N taps associated with the antenna to generateMN pulses in a receive signal vector x(k); arranging the pulses in athree-dimensional MNK matrix with K range cells; compressing the MNKmatrix to produce a compressed matrix, wherein the compressing isachieved at a low computational complexity that is linearly proportionalto MN; applying a progressive singular valued decomposition to thecompressed matrix to produce an inverse of a clutter covariance matrixC(k) from the inverse of the clutter covariance matrix obtained in aprevious time instant, and wherein the clutter covariance matrixobtained within a current time instant, wherein the PSVD provides a lowcomplexity way to compute the inverse of the clutter covariance matrixof the current time instant filtering the received signal vector x(k)according toz(k)=C(k)⁻¹ x(k); and detecting the target using z(k).